Mechanism Design With Money Christos Tzamos S CHOOL OF E LECTRICAL - - PowerPoint PPT Presentation

Mechanism Design With Money

Christos Tzamos

SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING NATIONAL TECHNICAL UNIVERSITY OF ATHENS, GREECE

Introduction VCG Mechanisms Combinatorial Auctions Motivation Game Theory Mechanism Design Impossibility Result

Motivation

Mechanism Design and Social Choice Design rules in order to make decisions based on people preferences when their interests are conflicting.

Christos Tzamos Mechanism Design With Money

Introduction VCG Mechanisms Combinatorial Auctions Motivation Game Theory Mechanism Design Impossibility Result

Game Theory

Game Theory Studies strategic situations where players choose different actions in an attempt to maximize their returns. Outcome Prediction - Solution Concepts Nash Equilibrium Pure Nash Equilibrium Dominant Strategy

Christos Tzamos Mechanism Design With Money

Introduction VCG Mechanisms Combinatorial Auctions Motivation Game Theory Mechanism Design Impossibility Result

Mechanism Design

Mechanism Design Mechanism design is the art of designing rules of a game to achieve a specific outcome under a certain solution concept. Social Choice as a Game A set A of different alternatives A set of n voters (the agents) N Each agent i has a linear order ≻i∈ L over the set A A function (mechanism) f : Ln → A that maps the agents’ preferences to a single alternative is called social choice function.

Christos Tzamos Mechanism Design With Money

Introduction VCG Mechanisms Combinatorial Auctions Motivation Game Theory Mechanism Design Impossibility Result

Properties

Onto ∀a ∈ A, ∃x ∈ Ln such that f(x) = a Unanimous if ∃a ∈ A such that ∀b ∈ A and i ∈ N, a ≻i b then f(≻1, . . . , ≻n) = a Pareto Optimal if f(≻1, . . . , ≻n) = a, then ∄b ∈ A such that b ≻i a, ∀i ∈ N

Christos Tzamos Mechanism Design With Money

Introduction VCG Mechanisms Combinatorial Auctions Motivation Game Theory Mechanism Design Impossibility Result

Properties - Incentive Compatibility

Strategic Manipulation by agent i ∃ ≻1, . . . , ≻n, ≻′

i∈ L such that b ≻i a where a = f(≻1, . . . , ≻i, . . . , ≻n)

and b = f(≻1, . . . , ≻′

i, . . . , ≻n).

Strategyproofness A social choice function is called incentive compatible or strategyproof or truthful if no agent can strategically manipulate it.

Christos Tzamos Mechanism Design With Money

Introduction VCG Mechanisms Combinatorial Auctions Motivation Game Theory Mechanism Design Impossibility Result

Impossibility Result

Gibbard-Satterthwaite Let f be an incentive compatible social choice function onto A, where |A| ≥ 3, then f is a dictatorship. Escape Routes Money Randomization Restricted domain of preferences

Christos Tzamos Mechanism Design With Money

Introduction VCG Mechanisms Combinatorial Auctions Setting and Outline Single Item Auction General Settings Weighted VCG

Setting and Outline

Measuring Preferences with Money Each agent has a value for every alternative a ∈ A, which is given by a private function vi : A → R where vi ∈ Vi. The value vi(a) corresponds to the amount of money agent i is willing to pay in order to force the outcome a. Extending the notion of a mechanism A mechanism is a social choice function f : V1 × V2 × · · · × Vn and a vector of payment functions p1, p2, . . . , pn, where pi : Vi → R is the amount that player i pays.

Christos Tzamos Mechanism Design With Money

Introduction VCG Mechanisms Combinatorial Auctions Setting and Outline Single Item Auction General Settings Weighted VCG

Desirable Properties

Incentive Compatibility Mechanism (f, p1, ..., pn) is incentive compatible if for each player i, every v1, ..., vn and every v′

i , we have that

vi(f(vi, v−i)) − pi(vi, v−i) ≥ vi(f(v′

i ,

v−i)) − pi(v′

i ,

v−i) Individual Rationality Mechanism (f, p1, . . . , pn) is individually rational if for each player i and every v1, ..., vn, we have that vi(f(v1, ..., vn)) − pi(vi, ..., vn)) ≥ 0 No Positive Transfers Mechanism (f, p1, ..., pn) has no positive transfers if for each player i and every v1, ..., vn, we have that pi(vi, ..., vn)) ≥ 0

Christos Tzamos Mechanism Design With Money

Introduction VCG Mechanisms Combinatorial Auctions Setting and Outline Single Item Auction General Settings Weighted VCG

Single Item Auction

Selling a single item The set of alternatives is the set of possible winners A = {i − wins|i ∈ N} The agent valuations are vi(i − wins) = wi and vi(j − wins) = 0∀j = i. The agent with the highest wi should get the item. How to choose payments No payments Player i should increase his bid to get the item. Winner pays bid In the case where the other agent valuations are lower the winner should decrease his bid Not incentive compatible

Christos Tzamos Mechanism Design With Money

Introduction VCG Mechanisms Combinatorial Auctions Setting and Outline Single Item Auction General Settings Weighted VCG

Single Item Auction - Truthful Mechanisms

Pay all agents Give the item to the highest bidder and pay everyone else the value

• f the winning bid.

Truthful but not efficient (positive transfers) Winner pays second highest bid - Vickrey Auction Give the item to the highest bidder i and make him pay p∗ = maxj=i wj. Everyone else doesn’t pay anything. Truthful and efficient (positive transfers)

Christos Tzamos Mechanism Design With Money

Introduction VCG Mechanisms Combinatorial Auctions Setting and Outline Single Item Auction General Settings Weighted VCG

General Setting

Maximizing Social Welfare There exists payment functions such that the mechanism that maximizes the social welfare i.e.

i∈N vi(a) is incentive compatible.

Vickrey-Clarke-Groves Mechanisms f(v1, ..., vn) ∈ argmaxa∈A

• i∈N vi(a)

pi(vi, v−i) = hi( v−i) −

j=i vj(f(v1, ..., vn))

where hi ∈ V−i → R arbitrary functions Theorem Every VCG mechanism is incentive compatible.

Christos Tzamos Mechanism Design With Money

Introduction VCG Mechanisms Combinatorial Auctions Setting and Outline Single Item Auction General Settings Weighted VCG

Choosing the hi’s

hi = 0 Extremely inefficient. The mechanism pays every player. The no positive transfers property is violated. Clarke Pivot Rule Choose each hi( v−i) = maxb∈A

• j=i vj(b). (=max social welfare

when player i doesn’t participate.) Player i is charged with the amount that the social welfare of the

• thers decreased due to his choice.

Theorem A VCG mechanism with the CPR makes no positive transfers.

Christos Tzamos Mechanism Design With Money

Introduction VCG Mechanisms Combinatorial Auctions Setting and Outline Single Item Auction General Settings Weighted VCG

Example

Summary & Example

a b c price utility I 1 5 10 5 10-5=5 II 7 6 4 4 III 9 8 7 7 sum 17 19 21 What if player I lies and says that her value for c is 7? a b c price utility I 1 5 10 7 2 5-2=3 II 7 6 4 III 9 8 7 sum 17 19 18 smaller than her utility when telling the truth!

Christos Tzamos Mechanism Design With Money

Introduction VCG Mechanisms Combinatorial Auctions Setting and Outline Single Item Auction General Settings Weighted VCG

Weighted VCG

Affine Maximizer A social choice function f is called affine maximizer if for some player weights w1, ..., wn ∈ R and some weights ca ∈ R for all a ∈ A, we have that f(v1, ..., vn) = argmaxa∈A(ca +

i∈N wivi(a))

Weighted VCG f(v1, ..., vn) = argmaxa∈A(ca +

i∈N wivi(a))

pi(vi, v−i) = hi( v−i) −

j=i(wj/wi)vj(f(v1, ..., vn)) − ca/wi

where hi ∈ V−i → R arbitrary functions Theorem (Roberts) If |A| ≥ 3, f is onto A, Vi = R|A|, then all incentive compatible mechanisms are Weighted VCG.

Christos Tzamos Mechanism Design With Money

Introduction VCG Mechanisms Combinatorial Auctions Setting and outline Single-Minded Case

Setting and outline

Problem Statement A set of m items M to be divided among the agents (A = (N ∪ {e})m) For each agent i, vi : 2M → R (monotone valuation function) Objective: Maximize social welfare Difficulties The allocation problem is NP-complete to compute optimally How to retrieve and represent agent valuations? (|A| is exponential)

Christos Tzamos Mechanism Design With Money

Introduction VCG Mechanisms Combinatorial Auctions Setting and outline Single-Minded Case

Setting and outline

Avoiding the difficulties Focus on simpler cases of valuation functions (linear, single minded) Introduce approximation VCG mechanisms VCG mechanisms require the computation of the optimal social

• welfare. They don’t work for approximations.

Christos Tzamos Mechanism Design With Money

Introduction VCG Mechanisms Combinatorial Auctions Setting and outline Single-Minded Case

Single-Minded Case

Each agent i is interested in obtaining a certain bundle of items Si. vi(T) = v∗

i , ∀T ⊇ Si

vi(T) = 0, ∀T ⊂ Si Proposition The allocation problem among single-minded bidders is NP-hard. We show this by a reduction to the INDEPENDENT-SET problem. The items are the edges and the bidders are the vertices. Each bidder’s bundle is the set of his adjacent edges with value v∗

i = 1.

Theorem No approximate mechanism exists with approximation ratio m1/2−ǫ.

Christos Tzamos Mechanism Design With Money

Introduction VCG Mechanisms Combinatorial Auctions Setting and outline Single-Minded Case

Single-Minded Case

Greedy Mechanism Order the agents bids by v∗

i /√Si decreasingly

Process agents in order and give them their desired bundle if available Theorem The greedy mechanism is incentive compatible and √m-approximate

Christos Tzamos Mechanism Design With Money

Introduction VCG Mechanisms Combinatorial Auctions Setting and outline Single-Minded Case

The End

Thank you!

Christos Tzamos Mechanism Design With Money